LMFDB - L-function 2-1900-95.94-c2-0-4

Dirichlet series

L(s) = 1

− 2.03·3^-s + 4.27i·7^-s − 4.86·9^-s + 5.47·11^-s − 0.634·13^-s − 1.69i·17^-s + (6.46 + 17.8i)19^-s − 8.70i·21^-s − 25.9i·23^-s + 28.1·27^-s + 48.8i·29^-s − 4.47i·31^-s − 11.1·33^-s − 6.12·37^-s + 1.28·39^-s + ⋯

L(s) = 1

− 0.677·3^-s + 0.611i·7^-s − 0.540·9^-s + 0.497·11^-s − 0.0487·13^-s − 0.0995i·17^-s + (0.340 + 0.940i)19^-s − 0.414i·21^-s − 1.12i·23^-s + 1.04·27^-s + 1.68i·29^-s − 0.144i·31^-s − 0.337·33^-s − 0.165·37^-s + 0.0330·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$-0.993 - 0.116i$
Analytic conductor:	$51.7712$
Root analytic conductor:	$7.19522$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1900} (949, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1900,\ (\ :1),\ -0.993 - 0.116i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.3493300968$
$L(\frac12)$	$\approx$	$0.3493300968$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 $
	19	$ 1 + (-6.46 - 17.8i)T $
good	3	$ 1 + 2.03T + 9T^{2} $
	7	$ 1 - 4.27iT - 49T^{2} $
	11	$ 1 - 5.47T + 121T^{2} $
	13	$ 1 + 0.634T + 169T^{2} $
	17	$ 1 + 1.69iT - 289T^{2} $
	23	$ 1 + 25.9iT - 529T^{2} $
	29	$ 1 - 48.8iT - 841T^{2} $
	31	$ 1 + 4.47iT - 961T^{2} $
	37	$ 1 + 6.12T + 1.36e3T^{2} $
	41	$ 1 + 16.9iT - 1.68e3T^{2} $
	43	$ 1 + 0.690iT - 1.84e3T^{2} $
	47	$ 1 - 23.0iT - 2.20e3T^{2} $
	53	$ 1 + 54.3T + 2.80e3T^{2} $
	59	$ 1 + 0.251iT - 3.48e3T^{2} $
	61	$ 1 - 39.5T + 3.72e3T^{2} $
	67	$ 1 + 96.3T + 4.48e3T^{2} $
	71	$ 1 - 16.0iT - 5.04e3T^{2} $
	73	$ 1 + 70.0iT - 5.32e3T^{2} $
	79	$ 1 + 74.6iT - 6.24e3T^{2} $
	83	$ 1 + 1.29iT - 6.88e3T^{2} $
	89	$ 1 - 141. iT - 7.92e3T^{2} $
	97	$ 1 + 125.T + 9.40e3T^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.252319751049635646030412662539, −8.738236372671934501763198371154, −7.896849313246852306204096438617, −6.85849859456424968701005484156, −6.15600928640243454735397199463, −5.47055103918082510618808153562, −4.71200702134801520013404860281, −3.55066593362792552672515706629, −2.57270640204902592334046766801, −1.29248444262865048689622535357, 0.11057045738831448442562735108, 1.21165959997715185052520654581, 2.62967589986792585515369734926, 3.70635176971753713405733415191, 4.61175403705811188250892053585, 5.48728448304277567570871236283, 6.22484953212382491683995015567, 7.01169147361004303300548623266, 7.77883376891597591359360330859, 8.689824768782208577310176529800

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.3493300968\)
\(L(\frac12)\)	\(\approx\)	\(0.3493300968\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more